Vectors in Component Form : Study with Video Lessons, Practice Problems & Examples

Understanding vectors involves representing them in component form, which includes identifying their x and y components. The magnitude of a vector can be calculated using the Pythagorean theorem, where the magnitude $|v|$ is given by $\sqrt{x^2+y^2}$. Additionally, vectors can be added or subtracted by combining their respective components. This foundational knowledge is crucial for further studies in mathematics and physics.

Position Vectors & Component Form

Welcome back, everyone. So up to this point, we've spent a lot of time talking about vectors and what they look like. Now recall that visually, vectors are these arrows drawn in space, and we've talked about how these arrows can be stretched, or shrink, or can be combined with other arrows to create resultant vectors. Well, what we're going to be talking about in this video is position vectors and component form. And this might sound a bit random and confusing, but don't sweat it because it turns out all we're really going to be learning about in this video is a simple way to write numbers that represent our vector. And I think you're going to find mathematically, it's actually very intuitive. So without further ado, let's get right into things because this is an important concept to understand.

Now let's say we have this vector, which we'll call vector $\mathbf{v}$. As you can see, this is an arrow drawn in space. Now what we can do is represent this as a position vector, and a position vector is simply a vector where the initial point is drawn at the origin. So if we want to draw vector $\mathbf{v}$ as a position vector, I just need to relocate this so the initial point is at the origin. And that right there is the position vector, and that's all there is to it. It's just moving your vector to the origin of the graph. Now the question becomes, how can we write this vector with some kind of numbers? Well, we can see that we have some numbers here on the graph, and we can use these numbers to figure out what our vector is. Because what we do is we represent these vectors using component form, where we have an x component and a y component. And all these components do is tell you the length of the vector in the x and y directions. So you can see in the x direction, we need to go 3 units to the right, so we'd have 3. And then in the y direction, we need to go 2 units up, so we'd have 2. So this vector is 3, 2, and that's all there is to it. As you can see, it's really straightforward.

Now it turns out that there are also ways that you can represent these vectors if you don't have a position vector. So say that you were given some initial point of the vector, like a point right there, and then a terminal point over here. You could figure out what the vector is in component form by using this equation down here. And to really put this equation to use and understand it rather than just looking at it, let's actually try an example where we have to do this. So in this example, we are told if a vector has an initial point (0.2, 0.3) and a terminal point (0.3, 0.5), without drawing the vector, write the vector in component form. So we're not allowed to just graph this immediately and figure out what it looks like. What we need to do is use this equation to figure out what our vector is.

But this equation is actually pretty simple to use. So all I need to do is recognize that our vector $\mathbf{v}$ is going to be the difference in the x components, the difference in the y components. So we can see that we have the final x minus the initial x, and then we're going to have the final y minus the initial y. Now I can see what these values are based on the points above. So if I go ahead and go to this first point, you see this first point is at (0.2, 0.3). I can see that the second point is at (0.3, 0.5). So I'm going to have for the x points, the difference between 0.3 and 0.2. So we're going to have $0.3-0.2$, and then we're going to subtract the y values. So the final y is 0.5, and then the initial y is 0.3. So this is what our vector is going to be. So we're going to have $0.3-0.2$ which is 0.1, and we are going to have $0.5-0.3$ which is 0.2. And that right there is the solution. That is vector $\mathbf{v}$.

So this is how you can solve problems when you can't initially draw them or don't initially have some kind of graph of the vector. Now if we want to know what this vector looks like, we actually can use this graph just for reference. Well, I can see that our vector is $⟨0.1,0.2⟩$. And if we draw this as a position vector, starting at the origin of our graph, we're going to go 0.1 to the right, and we're going to go 0.2 up. And that right there would be our vector $\mathbf{v}$. So as you can see, whatever you're dealing with these types of vectors, you're going to have the x component, which is how far we travel in the x direction, and the y component, which is how far we travel in the y direction. And that's always going to be the case when using component form.

So that is how you can represent vectors using numbers, and how you can draw position vectors which are at the origin of your graph. So, I hope you found this video helpful. Thanks for watching.

True or false: If $a ⃗=⟨3,2⟩$ and $b ⃗$ has initial point $(3,-1)$ & terminal point $(6,1)$, then $a ⃗=b ⃗$.

• A. True
• B. False
• C. Cannot be determined with given information

Answer: (A)

True or false: If $a ⃗=⟨3,-2⟩$ and $b ⃗$ has initial point $(-10,5)$ & terminal point $(-7,3)$, then $a ⃗=b ⃗$.

• A. True
• B. False
• C. Cannot be determined with the given information

Answer: (A)

Position Vectors & Component Form Example 1

Let's see if we can solve this example. So in this example, we're told if vector v has initial point (4,3) and terminal point (1,2), write vector v in component form and sketch its position vector. Now our first step is going to be to find vector v in component form, and we can do that using this equation. It's going to be x2-x1, y2-y1, and this will give us the vector we're looking for. Now the x2 is going to be the final x, or basically the terminal x, and I can see the terminal point here (1,2) is going to be that x value, so we're going to have 1-4. Next, we're going to have y2-y1. Now y2 is going to be the y value for the terminal point, which is 2, and then we're going to subtract the y value from the initial point, which is 3. So, this is the vector we end up getting. Now to solve this, what we can do is we can do the subtractions here. So we're going to have 1-4 which is negative 3, and we're going to have 2-3 which is negative 1. So the vector is -3, -1, and that is the vector that we're looking for in component form.

Now we're also asked to sketch the position vector, and the position vector is just going to be this vector that starts at the origin of the graph. So let's start here at the origin. I can see we're at negative 3, so we can go 3 units to the left, and then we're going to be at negative 1 which is 1 unit down, and this right here would be our vector v. Notice how it goes backward, it goes towards the negative side of the x-axis instead of the positive side because our vector is (-3,-1).

So this is how you can find the vector and sketch its corresponding position vector. I hope you found this video helpful. Thanks for watching.

Finding Magnitude of a Vector

Welcome back, everyone. So in the last video, we talked about how to write vectors using component form, which ends up looking something like this where we have these brackets. Now what we're going to be talking about in this video is how you can use this component form we learned about to find the magnitude of a vector. Now, the magnitude is something that we've talked about in previous videos. Recall that it describes how much of something you have, or basically, when it comes to vectors, it describes the total length of the vector itself. And it turns out finding this length, there's actually a pretty straightforward way you can calculate a number to represent it. So, without further ado, let's get right into an example of this, because this is an important concept to understand.

Now let's say that we have this vector down here, and we want to find the magnitude. Well, our first step should be to figure out what this vector is in component form. And we can do that by just taking a look at our graph. So I can see that we need to find the x and y components. Now the x component is going to be 4 units to the right, so our x component is 4, and our y component is going to be 3 units up. So our vector is 4, 3, and that's it. That's the vector in component form. But how could we use these components to figure out the length of this vector? Could you think of some sort of way that we could do that?

Well, I want you to recall something we learned about called the Pythagorean theorem. The Pythagorean theorem is an equation that we use on right triangles, and what it does is relates all the sides of a triangle together. So we have: a2 + b2 = c2 where c is the hypotenuse or long side of the triangle. Then you can rearrange this equation to get that c is equal to the square root of a2 + b2 And plugging in the two legs of the triangle will allow you to calculate the hypotenuse.

It turns out, we can actually use this exact same equation and the same strategy to find the magnitude of any vector. Notice something; these vectors here form a right triangle. These two vectors are perpendicular, and then this would just be the hypotenuse. So if we want to find the magnitude or length of this vector, all we need to do is take the square root of the x component squared and the y component squared, and add them together. Because that's the same thing as having these two legs of the triangle and treating them like the opposite and adjacent side. So let's go ahead and do that.

So we have this vector, magnitude that we're trying to calculate, and it's going to be the square root of our x component squared, which is 4 squared, plus our y component squared, which is 3 squared. Now 4 squared is 16, and 3 squared is 9. 16 plus 9 is 25, and the square root of 25 is 5. So this right here is the magnitude or length of our vector. It's 5 units long. And that is how you can calculate this using this Pythagorean theorem that we've already learned about.

So you can just use this equation whenever you have your vector in component form. Now, to make sure we understand this well, let's actually try an example where there's not a nice graph given to us. So in this example, we're asked to calculate the magnitude of vector pq if we're given the initial point or some point p, 12, and the final point, 53. Now to figure this out, recall that we learned this equation in the previous video, which allows us to calculate our vector in component form if we're given 2 points.

Now this is going to be the final point, which we'll call point 2, and this is going to be the initial point, point 1. So let's go ahead and use this equation. We have that our vector v is going to be equal to the final x minus the initial x, comma the final y minus the initial y. Now what I'm going to do is plug the numbers in. So we're going to have 5 and 1, which are the initial x components. So we'll have 5 minus 1, and that's going to be comma, and then we're going to have subtracting the y components. So we're going to have 3 minus 2. So this is what our vector is going to be. Now 5 minus 1 is 4, and 3 minus 2 is 1. So this right here is our vector in component form.

Now since we've calculated the vector in component form, recall that we can now use this equation to figure out what the magnitude of our vector is. So the magnitude of our vector is going to be the x component squared plus the y component squared, and all of that is going to be underneath a square root. So let's go ahead and plug the values in. So we're going to have the square root of our x component squared, which is 4, plus our y component squared, which is 1. Now 4 squared, that comes out to 16, and 1 squared is just 1. 16 plus 1 is 17, and we can't simplify the square root down any further. So the magnitude of our vector is 17, and that is the solution to this problem. So this is how you can find the magnitude of any vector. And sometimes you will be given the vector in component form, other times you'll need to figure out what the vector is in component form. You could always use this equation when you're trying to calculate it.

Hope you found this video helpful. Thanks for watching.

If vector $v ⃗=⟨-4,-10⟩$, calculate the magnitude $|v ⃗ |$.

• A. $4\surd29$
• B. $2\surd116$
• C. $2√29$
• D. $\surd-116$

Answer: (C)

If vector $v ⃗$ has initial point $(-1,2)$ and terminal point $(9,5)$, calculate the magnitude $|v ⃗ |$.

• A. $2√29$
• B. $\surd109$
• C. $\surd73$
• D. $\surd113$

Answer: (B)

Finding Magnitude of a Vector Example 1

Let's give this problem a try. So in this problem, we're told if vector v has an initial point (1,2) and a terminal point (4,4), sketch v as a position vector and calculate its magnitude. Now, to solve this problem, the first thing I'm going to do is figure out what vector v is in component form. This is always a good first step when you're dealing with these types of problems where you're given the initial and terminal point of a vector. Now to solve this, to find this component form, what I can do is take the final x and subtract off the initial x, and then take the final y and subtract off the initial y, and that's going to give us our vector. Now, first off, we can see that for x₂, this is going to be the x value for the terminal point, which is 4. So what I can first do is plug in 4 into this equation. Now, next, I'm going to subtract off the initial x, and the initial x is 1. So we're going to have 4 minus 1 for the difference in the x's. Now as for the y's, I can see that we have y₂. And y₂ is going to be this y value for the terminal point, which is 4, and then this is going to be minus the initial y, and the initial y is 2. So we're going to have 4 minus 2. So this right here gives us the vector (3,2).

Now that we have our vector in component form, what we can do is now sketch the position vector. The position vector just means that our vector is going to start at the origin. So if I go here at the origin of our graph, I can see that our vector is (3,2). So that means on the x-axis, we're going to move over 3 units, then on the y-axis, we're going to move up 2 units. This is what our vector is going to look like, and that's vector v sketched as a position vector.

We've now found the component form and have sketched our position vector. Our last step is going to be to calculate the magnitude of this vector. And to calculate the magnitude, we can use this equation: Magnitude = Vx2 + Vy2 , where Vx is the x component and Vy is the y component. So we can see that we have these values that we calculated already, so we're going to have the Vx which is 3, so we have 32 plus 22. Now 32 is 9, so we're going to have the square root of 9+4 which is 13. So the magnitude of our vector 'v' is equal to the square root of 13. And this is the magnitude of our vector, as well as our vector sketched as a position vector, and that is the solution to this problem. So I hope you found this video helpful. Thanks for watching.

Algebraic Operations on Vectors

Welcome back, everyone. So, in previous videos, we've talked about how you can add or subtract vectors using the tip-to-tail method. Recall if we had one vector, we could add the tip of that vector to the tail of a second vector, and the resultant vector that connected the two points would give you the sum of those vectors. This should be reviewed from previous videos. Now it turns out there is also a way that you can add or subtract vectors if you are given numbers as opposed to just a graph. And that's what we're going to be talking about in this video. This is a very important skill to have in this course as well as likely future math or science courses, so let's just go ahead and get right into things.

Now if you have 2 vectors and you're given them in this component form, the way that you can add or subtract these vectors is by adding or subtracting the individual x and y components. So to understand this, let's say that I have vector v, and I want to add or subtract it to vector u. Well, what I just need to do is add or subtract the individual components. So I can add or subtract the x components, so we'll have $v_x\pm u_x$. And then, likewise, I can add or subtract the individual y components. And that's all there really is to adding or subtracting vectors.

So let's go ahead and try it with this example down here. So here we have vector (2, 3), and we wish to add it to vector (3, -1). Well, to do this, I'm first going to add the x components, which are 2 and 3, and then I'm going to add the y components, which are 3 and -1. $2+3$ is 5, and $3+(-1)$ is positive 2. So this right here is the vector $v+u$. That's all there is to it. That's the solution. Now it turns out if you actually use the tip-to-tail method on these vectors, you're going to find that you actually will get this result.

So let's say I have vector (2,3), which I can draw on this graph right here. And then let's also say I have this vector (3, -1). So I can draw this right about there. So we have vector v, and here we have vector u. And if I go ahead and use the tip-to-tail method where I take the initial point of this vector, of the first vector, and connect it to the terminal point of the second vector, notice how this is the vector that we get for $v+u$. But notice how this vector ended up being 1, 2, 3, 4, 5 units to the right, and it ended up being 1, 2 units up. So we get the exact same result if we use this tip-to-tail method. So as you can see the math actually checks out here when we do this visually.

Now, something else I do want to touch on is this idea of multiplying a vector by a scalar, which is something else we've discussed. You can do this with the numbers too, but I think the best way to understand this is to try an example where we have to do this. So let's say we have this situation where we have 2 vectors, v and u. And we're asked to find the vector $v-3u$. Now to solve this, what I'm first going to do is write out what we have. So you can see that we have vector v, and that's given to us. It's (8, 5). Now what I also see is that we have vector u, but what I need to do is figure out what vector 3u is. So to find vector 3u, well, that means I need to take our vector u, which is (2, 4), and I need to multiply it by 3. But how exactly do I do this? Well, it turns out if you want to multiply a vector by a scalar, all you need to do is distribute the scalar into each of the individual x and y components. So let's say that we have some scalar constant k. All I need to do is multiply this by the x and the y components. So we'll have $k\times$ the x component and then $k\times$ the y component. So going down here, I can just distribute this 3 into each component. So we're going to end up with $3\times 2$ is 6, and $3\times 4$ is 12. So this is the vector 3u. So we have vector 3u and we have vector v, but all I need to do from here is figure out what vector $v-3u$ is. And to find $v-3u$, well, I just figured out what v and 3u are, so I just subtract the two vectors. We said that v is (8, 5), and we said that 3u is (6, 12), so I just need to subtract them. So we'll have $8-6$ which comes out to 2, and then we're going to have $5-12$ which comes out to negative 7. So this here is the solution to the problem, and this is how you can do these basic operations with vectors like adding or subtracting vectors, or multiplying vectors by scalars when you're dealing with numbers rather than a graph. So, I hope you found this video helpful. Thanks for watching.

If vectors $v ⃗=⟨2,1⟩$, $u ⃗=⟨3,4⟩$, and $w ⃗=⟨-1,1⟩$, calculate $v ⃗+u ⃗-w ⃗$.

• A. $\langle4,4\rangle$
• B. $\langle5,4\rangle$
• C. $\langle6,6\rangle$
• D. $\langle6,4\rangle$

Answer: (D)

If vectors $v ⃗=⟨4,1⟩$, $u ⃗=⟨-8,3⟩$, and $w ⃗=⟨-2,-1⟩$, calculate $w ⃗-3(v ⃗+u ⃗)$.

• A. $\langle10,13\rangle$
• B. $\langle-14,13\rangle$
• C. $\langle-14,-13\rangle$
• D. $\langle10,-13\rangle$

Answer: (D)

Algebraic Operations on Vectors Example 1

Let's see if we can solve this example. So in this example, we're told if vector a equals 31, and vector b equals negative 49, and vector c is equal to 5 times vector a minus 2 times vector b, we're asked to calculate the magnitude of vector c. Now, what I can see here is that we're trying to ultimately find the magnitude of this vector, but this vector depends on the other two vectors we have in this problem. So we definitely have our work cut out for us here. But whenever solving these problems where we have multiple vector operations, I personally like to start small with the problem and then work my way to solving the whole thing. So let's actually do that.

Now the first thing I'm going to do is figure out what 5 times vector a is, because I'm going really be focusing on this vector c here. And to find 5 times a, well 5 times vector a is just going to be 5 times this vector. So we're going to have 5 times 31. Now, 5 times 31 we can multiply this scalar into each of the components, so we'll have 5 times 31. This is the vector 5a. Next, I'm going to figure out what vector 2b is. So vector 2b is going to be 2 times vector b. Well, I can see here that vector b is negative 49, so that's going to be our vector, and then what I need to do is distribute the scalar into this vector as well. So we're going to have 2 times negative 49. This is vector 2b.

What I need to do from here, they need to subtract these 2 vectors. To find vector c, it's going to be 5a minus 2b. So we can see here that 5a is 155, and then we can see that vector 2b is negative 98. And then what I can do is subtract these 2 vectors. If I subtract them we're going to have 155 minus negative 98, and 155 minus negative 98 is the same thing as 155 + 98, which is 253. So this right here is vector c, 253. And now that I've found vector c, our last step is to just find the magnitude of c.

We can use the Pythagorean theorem, and the Pythagorean theorem for these vectors is going to be the square root of the x component of c squared. Now, 253 squared turns out to be 64,009. So the magnitude of C is the square root of 64,009, and that is the solution to this problem. So this is how you can handle multiple vector operations, as well as finding the magnitude of a vector once you've already done operations on that vector. I hope you found this video helpful. Thanks for watching.

Here’s what students ask on this topic:

What is the component form of a vector?

The component form of a vector is a way to represent the vector using its horizontal (x) and vertical (y) components. If a vector starts at the origin and ends at the point (x, y), its component form is written as $(x,y)$. These components indicate how far the vector moves in the x and y directions, respectively. For example, a vector that moves 3 units to the right and 2 units up would be represented as $(3,2)$.

How do you find the magnitude of a vector in component form?

To find the magnitude of a vector in component form, you use the Pythagorean theorem. If the vector is represented as $(x,y)$, the magnitude $\left|v\right|$ is calculated using the formula $\sqrt{x^2+y^2}$. For example, if the vector is $(4,3)$, the magnitude is $\sqrt{4\text{msup}+3\text{msup}}$, which simplifies to $\sqrt{25}$ or 5.

How do you add or subtract vectors in component form?

To add or subtract vectors in component form, you simply add or subtract their corresponding components. If you have vectors $(x,y)$ and $(u,v)$, their sum is $(x+u,y+v)$ and their difference is $(x-u,y-v)$. For example, adding vectors $(2,3)$ and $(3,-1)$ results in $(2+3,3+-1)$, which simplifies to $(5,2)$.

How do you multiply a vector by a scalar in component form?

To multiply a vector by a scalar, you multiply each component of the vector by the scalar. If the vector is $(x,y)$ and the scalar is $k$, the resulting vector is $(\mathrm{kx},\mathrm{ky})$. For example, if the vector is $(2,4)$ and the scalar is 3, the resulting vector is $(3\times 2,3\times 4)$, which simplifies to $(6,12)$.

How do you find the component form of a vector given its initial and terminal points?

To find the component form of a vector given its initial point $(x1,y1)$ and terminal point $(x2,y2)$, you subtract the coordinates of the initial point from the coordinates of the terminal point. The component form is $(x2-x1,y2-y1)$. For example, if the initial point is $(2,3)$ and the terminal point is $(5,7)$, the component form is $(5-2,7-3)$, which simplifies to $(3,4)$.